The strong form of the initial boundary value problem (IBVP) can be stated as follows
[1]}, [2]}, [3]}, [4]}:
given the fluid discharge \(q_s:\cup _{i=1}^{n_f}\Gamma _i\times \mathcal {T}\rightarrow \mathbb {R}\) , the prescribed boundary displacement \(\overline{u}:\partial \Omega _u\times \mathcal {T}\rightarrow \mathbb {R}^3\)  and traction \(\overline{t}:\partial \Omega _\sigma \times \mathcal {T}\rightarrow \mathbb {R}^3\) , the prescribed fracture boundary pressure \(\overline{p}:\cup _{i=1}^{n_f}\partial \Gamma _{i,p}\times \mathcal {T}\rightarrow \mathbb {R}\)  and flux \(\overline{q}:\cup _{i=1}^{n_f}\partial \Gamma _{i,q}\times \mathcal {T}\rightarrow \mathbb {R}\) , the initial displacement \(u_0:\overline{\Omega }\rightarrow \mathbb {R}^3\)  and pressure \(p_0:\cup _{i=1}^{n_f}\overline{\Gamma }_i\rightarrow \mathbb {R}\) ,
find the displacement \(u:\overline{\Omega }\times [0,t_{\max }]\rightarrow \mathbb {R}^3\) , the traction \(t:\cup _{i=1}^{n_f} \Gamma _i\times [0,t_{\max }]\rightarrow \mathbb {R}^3\) , and pressure \(p:\cup _{i=1}^{n_f} \overline{\Gamma }_i\times [0,t_{\max }]\rightarrow \mathbb {R}\)  such that:
\(- \nabla \cdot \sigma  (u) &= 0 & \mbox{ in }& \Omega \times \mathcal {T}&&& &\mbox{(linear momentum balance)}, \\\dot{g}_N(u) + \nabla \cdot q (u,p) &= q_s & \mbox{ in }&\cup _{i=1}^{n_f}\Gamma _i \times \mathcal {T} &&& &\mbox{(mass balance)}, \\t - p n_i &= 0 & \mbox{ on }& \cup _{i=1}^{n_f}\Gamma _i \times \mathcal {T}&&& &\mbox{(traction balance)}, \\\multicolumn{2}{l}{\text{respecting the boundary conditions}}\\u &= \bar{u} & \mbox{ on }& \partial \Omega _{u} \times \mathcal {T} &&& &\mbox{(prescribedboundary displacement)}, \\\sigma (u) \cdot n_\Omega &= \bar{t} & \mbox{ on }&\partial \Omega _{\sigma } \times \mathcal {T} &&& &\mbox{(prescribed boundary traction)},\\p &= \bar{p} & \mbox{ on }& \cup _{i=1}^{n_f}\partial \Gamma _{i,p} \times \mathcal {T} &&& &\mbox{(prescribed boundarypressure)}, \\q(u,p) \cdot m_i &= \bar{q} & \mbox{ on }& \cup _{i=1}^{n_f}\partial \Gamma _{i,q} \times \mathcal {T} &&&&\mbox{(prescribed boundary flux)}, \\\multicolumn{2}{l}{\text{and initial conditions}}\\u|_{t=0} &= u_0 & \mbox{ in }& \overline{\Omega } &&& &\mbox{(initialdisplacement)}, \\p|_{t=0} &= p_0 & \mbox{ in }& \cup _{i=1}^{n_f}\overline{\Gamma }_i &&& &\mbox{(initial pressure)},\\\multicolumn{2}{l}{\text{subject to the constraints over each $\Gamma _i$ and for every time in $\mathcal {T}$}}\\t_N = t \cdot n_i &\le 0, & g_N = \llbracket u \rrbracket \cdot n_i \ge 0, \quad t_N g_N &= 0 &&& &\mbox{(normal contact conditions)}, \\\left\Vert  t_T \right\Vert _2 - \tau _{\max }(t_N) & \le 0, & \dot{g}_T \cdot t_T - \tau _{\max }(t_N) \left\Vert  \dot{g}_T \right\Vert _2 &= 0 &&& &\mbox{(Coulomb frictional law)}. \) 
